Integrand size = 231, antiderivative size = 76 \[ \int \frac {2 \sqrt {3} x \left (-8 \sqrt {6}+16 \sqrt {2-\sqrt {3}}+8 \sqrt {3 \left (2-\sqrt {3}\right )}-8 \sqrt {2 \left (2-\sqrt {3}\right )} x^2-6 \sqrt {2} x^4+2 \sqrt {6} x^4+12 \sqrt {2-\sqrt {3}} x^4+4 \sqrt {3 \left (2-\sqrt {3}\right )} x^4-4 \sqrt {2 \left (2-\sqrt {3}\right )} x^6+\sqrt {2-\sqrt {3}} x^8\right )}{\left (2+2 \sqrt {3}-2 \sqrt {2} x^2+x^4\right )^2 \left (-4 \sqrt {3}+4 \sqrt {3 \left (2-\sqrt {3}\right )} x^2-3 x^4+\sqrt {3} x^4\right )} \, dx=\frac {24-12 \sqrt {3}-6 \sqrt {2 \left (7-4 \sqrt {3}\right )} x^2}{24-24 \sqrt {3}+24 \sqrt {2 \left (7-4 \sqrt {3}\right )} x^2-24 x^4+12 \sqrt {3} x^4} \] Output:
(24-12*3^(1/2)-6*(2*2^(1/2)-6^(1/2))*x^2)/(24-24*3^(1/2)+24*(2*2^(1/2)-6^( 1/2))*x^2-24*x^4+12*3^(1/2)*x^4)
Leaf count is larger than twice the leaf count of optimal. \(474\) vs. \(2(76)=152\).
Time = 1.84 (sec) , antiderivative size = 474, normalized size of antiderivative = 6.24 \[ \int \frac {2 \sqrt {3} x \left (-8 \sqrt {6}+16 \sqrt {2-\sqrt {3}}+8 \sqrt {3 \left (2-\sqrt {3}\right )}-8 \sqrt {2 \left (2-\sqrt {3}\right )} x^2-6 \sqrt {2} x^4+2 \sqrt {6} x^4+12 \sqrt {2-\sqrt {3}} x^4+4 \sqrt {3 \left (2-\sqrt {3}\right )} x^4-4 \sqrt {2 \left (2-\sqrt {3}\right )} x^6+\sqrt {2-\sqrt {3}} x^8\right )}{\left (2+2 \sqrt {3}-2 \sqrt {2} x^2+x^4\right )^2 \left (-4 \sqrt {3}+4 \sqrt {3 \left (2-\sqrt {3}\right )} x^2-3 x^4+\sqrt {3} x^4\right )} \, dx=\frac {\sqrt {3} \left (-1148625788556528844066159033262220068722206008007247900413395522084959231874596984923161348+663159408221258093117811630391460877747964543360199893206544595130744270586461480035682196 \sqrt {3}+242733190167635375474169843148282636230314782440022122004181389908674938370357203402689524 \sqrt {12-6 \sqrt {3}}-420426218053622717643631246206799161400925575248598068524055544875096370533184512002096630 \sqrt {4-2 \sqrt {3}}+\left (3889612146288378889767408303411449425825709236643908955354495069395079215720573720316532049 \sqrt {2}-2245668619702833580465139423609379981824461101336801162508724266862245809945670735146776573 \sqrt {6}-1643943526585545309302265021514972484744442185423605911246526728964400863501192436128705424 \sqrt {6-3 \sqrt {3}}+2847393712820121851628003284667408398787755831578416711092614144413117794707229269533951680 \sqrt {2-\sqrt {3}}\right ) x^2\right )}{\left (-322087496685481418858085119+185957302913975393256650292 \sqrt {3}+68065085482369716172535224 \sqrt {12-6 \sqrt {3}}-117892186276983133050206984 \sqrt {4-2 \sqrt {3}}\right ) \left (-252086908569103685779+145542444521551451775 \sqrt {3}+53272073800473568424 \sqrt {12-6 \sqrt {3}}-92269938446981769376 \sqrt {4-2 \sqrt {3}}\right ) \left (-5651986252505+3263175794250 \sqrt {3}+1192503998950 \sqrt {12-6 \sqrt {3}}-2065477490966 \sqrt {4-2 \sqrt {3}}\right ) \left (-662849748+382696633 \sqrt {3}+137980318 \sqrt {12-6 \sqrt {3}}-238988562 \sqrt {4-2 \sqrt {3}}\right ) \left (-3194627+1844361 \sqrt {3}+659824 \sqrt {12-6 \sqrt {3}}-1142984 \sqrt {4-2 \sqrt {3}}\right ) \left (-425692+245853 \sqrt {3}+84452 \sqrt {12-6 \sqrt {3}}-146092 \sqrt {4-2 \sqrt {3}}\right ) \left (1702768-983412 \sqrt {3}-337808 \sqrt {12-6 \sqrt {3}}+584368 \sqrt {4-2 \sqrt {3}}-4 \left (-251227 \sqrt {2}+145017 \sqrt {6}+90901 \sqrt {6-3 \sqrt {3}}-157580 \sqrt {2-\sqrt {3}}\right ) x^2+\left (-3194627+1844361 \sqrt {3}+659824 \sqrt {12-6 \sqrt {3}}-1142984 \sqrt {4-2 \sqrt {3}}\right ) x^4\right )} \] Input:
Integrate[(2*Sqrt[3]*x*(-8*Sqrt[6] + 16*Sqrt[2 - Sqrt[3]] + 8*Sqrt[3*(2 - Sqrt[3])] - 8*Sqrt[2*(2 - Sqrt[3])]*x^2 - 6*Sqrt[2]*x^4 + 2*Sqrt[6]*x^4 + 12*Sqrt[2 - Sqrt[3]]*x^4 + 4*Sqrt[3*(2 - Sqrt[3])]*x^4 - 4*Sqrt[2*(2 - Sqr t[3])]*x^6 + Sqrt[2 - Sqrt[3]]*x^8))/((2 + 2*Sqrt[3] - 2*Sqrt[2]*x^2 + x^4 )^2*(-4*Sqrt[3] + 4*Sqrt[3*(2 - Sqrt[3])]*x^2 - 3*x^4 + Sqrt[3]*x^4)),x]
Output:
(Sqrt[3]*(-114862578855652884406615903326222006872220600800724790041339552 2084959231874596984923161348 + 6631594082212580931178116303914608777479645 43360199893206544595130744270586461480035682196*Sqrt[3] + 2427331901676353 75474169843148282636230314782440022122004181389908674938370357203402689524 *Sqrt[12 - 6*Sqrt[3]] - 42042621805362271764363124620679916140092557524859 8068524055544875096370533184512002096630*Sqrt[4 - 2*Sqrt[3]] + (3889612146 28837888976740830341144942582570923664390895535449506939507921572057372031 6532049*Sqrt[2] - 22456686197028335804651394236093799818244611013368011625 08724266862245809945670735146776573*Sqrt[6] - 1643943526585545309302265021 514972484744442185423605911246526728964400863501192436128705424*Sqrt[6 - 3 *Sqrt[3]] + 28473937128201218516280032846674083987877558315784167110926141 44413117794707229269533951680*Sqrt[2 - Sqrt[3]])*x^2))/((-3220874966854814 18858085119 + 185957302913975393256650292*Sqrt[3] + 6806508548236971617253 5224*Sqrt[12 - 6*Sqrt[3]] - 117892186276983133050206984*Sqrt[4 - 2*Sqrt[3] ])*(-252086908569103685779 + 145542444521551451775*Sqrt[3] + 5327207380047 3568424*Sqrt[12 - 6*Sqrt[3]] - 92269938446981769376*Sqrt[4 - 2*Sqrt[3]])*( -5651986252505 + 3263175794250*Sqrt[3] + 1192503998950*Sqrt[12 - 6*Sqrt[3] ] - 2065477490966*Sqrt[4 - 2*Sqrt[3]])*(-662849748 + 382696633*Sqrt[3] + 1 37980318*Sqrt[12 - 6*Sqrt[3]] - 238988562*Sqrt[4 - 2*Sqrt[3]])*(-3194627 + 1844361*Sqrt[3] + 659824*Sqrt[12 - 6*Sqrt[3]] - 1142984*Sqrt[4 - 2*Sqr...
Leaf count is larger than twice the leaf count of optimal. \(216\) vs. \(2(76)=152\).
Time = 3.62 (sec) , antiderivative size = 216, normalized size of antiderivative = 2.84, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {6, 6, 6, 6, 27, 7266, 2126, 2191, 27, 2191, 27}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {2 \sqrt {3} x \left (\sqrt {2-\sqrt {3}} x^8-4 \sqrt {2 \left (2-\sqrt {3}\right )} x^6+4 \sqrt {3 \left (2-\sqrt {3}\right )} x^4+12 \sqrt {2-\sqrt {3}} x^4+2 \sqrt {6} x^4-6 \sqrt {2} x^4-8 \sqrt {2 \left (2-\sqrt {3}\right )} x^2+8 \sqrt {3 \left (2-\sqrt {3}\right )}+16 \sqrt {2-\sqrt {3}}-8 \sqrt {6}\right )}{\left (x^4-2 \sqrt {2} x^2+2 \sqrt {3}+2\right )^2 \left (\sqrt {3} x^4-3 x^4+4 \sqrt {3 \left (2-\sqrt {3}\right )} x^2-4 \sqrt {3}\right )} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {2 \sqrt {3} x \left (\sqrt {2-\sqrt {3}} x^8-4 \sqrt {2 \left (2-\sqrt {3}\right )} x^6+4 \sqrt {3 \left (2-\sqrt {3}\right )} x^4+12 \sqrt {2-\sqrt {3}} x^4+2 \sqrt {6} x^4-6 \sqrt {2} x^4-8 \sqrt {2 \left (2-\sqrt {3}\right )} x^2+8 \sqrt {3 \left (2-\sqrt {3}\right )}+16 \sqrt {2-\sqrt {3}}-8 \sqrt {6}\right )}{\left (x^4-2 \sqrt {2} x^2+2 \sqrt {3}+2\right )^2 \left (\left (\sqrt {3}-3\right ) x^4+4 \sqrt {3 \left (2-\sqrt {3}\right )} x^2-4 \sqrt {3}\right )}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {2 \sqrt {3} x \left (\sqrt {2-\sqrt {3}} x^8-4 \sqrt {2 \left (2-\sqrt {3}\right )} x^6+\left (2 \sqrt {6}-6 \sqrt {2}\right ) x^4+4 \sqrt {3 \left (2-\sqrt {3}\right )} x^4+12 \sqrt {2-\sqrt {3}} x^4-8 \sqrt {2 \left (2-\sqrt {3}\right )} x^2+8 \sqrt {3 \left (2-\sqrt {3}\right )}+16 \sqrt {2-\sqrt {3}}-8 \sqrt {6}\right )}{\left (x^4-2 \sqrt {2} x^2+2 \sqrt {3}+2\right )^2 \left (\left (\sqrt {3}-3\right ) x^4+4 \sqrt {3 \left (2-\sqrt {3}\right )} x^2-4 \sqrt {3}\right )}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {2 \sqrt {3} x \left (\sqrt {2-\sqrt {3}} x^8-4 \sqrt {2 \left (2-\sqrt {3}\right )} x^6+\left (12 \sqrt {2-\sqrt {3}}+4 \sqrt {3 \left (2-\sqrt {3}\right )}\right ) x^4+\left (2 \sqrt {6}-6 \sqrt {2}\right ) x^4-8 \sqrt {2 \left (2-\sqrt {3}\right )} x^2+8 \sqrt {3 \left (2-\sqrt {3}\right )}+16 \sqrt {2-\sqrt {3}}-8 \sqrt {6}\right )}{\left (x^4-2 \sqrt {2} x^2+2 \sqrt {3}+2\right )^2 \left (\left (\sqrt {3}-3\right ) x^4+4 \sqrt {3 \left (2-\sqrt {3}\right )} x^2-4 \sqrt {3}\right )}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {2 \sqrt {3} x \left (\sqrt {2-\sqrt {3}} x^8-4 \sqrt {2 \left (2-\sqrt {3}\right )} x^6+\left (-6 \sqrt {2}+2 \sqrt {6}+12 \sqrt {2-\sqrt {3}}+4 \sqrt {3 \left (2-\sqrt {3}\right )}\right ) x^4-8 \sqrt {2 \left (2-\sqrt {3}\right )} x^2+8 \sqrt {3 \left (2-\sqrt {3}\right )}+16 \sqrt {2-\sqrt {3}}-8 \sqrt {6}\right )}{\left (x^4-2 \sqrt {2} x^2+2 \sqrt {3}+2\right )^2 \left (\left (\sqrt {3}-3\right ) x^4+4 \sqrt {3 \left (2-\sqrt {3}\right )} x^2-4 \sqrt {3}\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \sqrt {3} \int \frac {x \left (-\sqrt {2-\sqrt {3}} x^8+4 \sqrt {2 \left (2-\sqrt {3}\right )} x^6+2 \sqrt {2} \left (3-\sqrt {3}-\sqrt {12-6 \sqrt {3}}-3 \sqrt {4-2 \sqrt {3}}\right ) x^4+8 \sqrt {2 \left (2-\sqrt {3}\right )} x^2+8 \left (\sqrt {6}-\sqrt {6-3 \sqrt {3}}-2 \sqrt {2-\sqrt {3}}\right )\right )}{\left (x^4-2 \sqrt {2} x^2+2 \left (1+\sqrt {3}\right )\right )^2 \left (\left (3-\sqrt {3}\right ) x^4-4 \sqrt {3 \left (2-\sqrt {3}\right )} x^2+4 \sqrt {3}\right )}dx\) |
| \(\Big \downarrow \) 7266 |
| \(\displaystyle \sqrt {3} \int \frac {-\sqrt {2-\sqrt {3}} x^8+4 \sqrt {2 \left (2-\sqrt {3}\right )} x^6+2 \sqrt {2} \left (3-\sqrt {3}-\sqrt {12-6 \sqrt {3}}-3 \sqrt {4-2 \sqrt {3}}\right ) x^4+8 \sqrt {2 \left (2-\sqrt {3}\right )} x^2+8 \left (\sqrt {6}-\sqrt {6-3 \sqrt {3}}-2 \sqrt {2-\sqrt {3}}\right )}{\left (x^4-2 \sqrt {2} x^2+2 \left (1+\sqrt {3}\right )\right )^2 \left (\left (3-\sqrt {3}\right ) x^4-4 \sqrt {3 \left (2-\sqrt {3}\right )} x^2+4 \sqrt {3}\right )}dx^2\) |
| \(\Big \downarrow \) 2126 |
| \(\displaystyle \frac {\sqrt {3} \int \frac {-\sqrt {2-\sqrt {3}} x^8+4 \sqrt {2 \left (2-\sqrt {3}\right )} x^6+2 \sqrt {2} \left (3-\sqrt {3}-\sqrt {12-6 \sqrt {3}}-3 \sqrt {4-2 \sqrt {3}}\right ) x^4+8 \sqrt {2 \left (2-\sqrt {3}\right )} x^2+8 \left (\sqrt {6}-\sqrt {6-3 \sqrt {3}}-2 \sqrt {2-\sqrt {3}}\right )}{\left (x^4-2 \sqrt {2} x^2+2 \left (1+\sqrt {3}\right )\right )^3}dx^2}{3-\sqrt {3}}\) |
| \(\Big \downarrow \) 2191 |
| \(\displaystyle \frac {\sqrt {3} \left (\frac {\int -\frac {16 \left (\sqrt {3 \left (2-\sqrt {3}\right )} x^4-2 \sqrt {6 \left (2-\sqrt {3}\right )} x^2-3 \left (\sqrt {2}+\sqrt {6}\right )+6 \sqrt {2-\sqrt {3}}+8 \sqrt {6-3 \sqrt {3}}\right )}{\left (x^4-2 \sqrt {2} x^2+2 \left (1+\sqrt {3}\right )\right )^2}dx^2}{16 \sqrt {3}}+\frac {\left (3 \sqrt {2}-\sqrt {6}-2 \sqrt {6-3 \sqrt {3}}\right ) x^2+2 \left (3 \sqrt {4-2 \sqrt {3}}-\sqrt {3} \left (2-\sqrt {4-2 \sqrt {3}}\right )\right )}{\sqrt {3} \left (x^4-2 \sqrt {2} x^2+2 \left (1+\sqrt {3}\right )\right )^2}\right )}{3-\sqrt {3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {3} \left (\frac {\left (3 \sqrt {2}-\sqrt {6}-2 \sqrt {6-3 \sqrt {3}}\right ) x^2+2 \left (3 \sqrt {4-2 \sqrt {3}}-\sqrt {3} \left (2-\sqrt {4-2 \sqrt {3}}\right )\right )}{\sqrt {3} \left (x^4-2 \sqrt {2} x^2+2 \left (1+\sqrt {3}\right )\right )^2}-\frac {\int \frac {\sqrt {3 \left (2-\sqrt {3}\right )} x^4-2 \sqrt {6 \left (2-\sqrt {3}\right )} x^2-3 \left (\sqrt {2}+\sqrt {6}\right )+6 \sqrt {2-\sqrt {3}}+8 \sqrt {6-3 \sqrt {3}}}{\left (x^4-2 \sqrt {2} x^2+2 \left (1+\sqrt {3}\right )\right )^2}dx^2}{\sqrt {3}}\right )}{3-\sqrt {3}}\) |
| \(\Big \downarrow \) 2191 |
| \(\displaystyle \frac {\sqrt {3} \left (\frac {\left (3 \sqrt {2}-\sqrt {6}-2 \sqrt {6-3 \sqrt {3}}\right ) x^2+2 \left (3 \sqrt {4-2 \sqrt {3}}-\sqrt {3} \left (2-\sqrt {4-2 \sqrt {3}}\right )\right )}{\sqrt {3} \left (x^4-2 \sqrt {2} x^2+2 \left (1+\sqrt {3}\right )\right )^2}-\frac {\frac {\int -\frac {6 \sqrt {2} \left (1+\sqrt {3}-\sqrt {12-6 \sqrt {3}}-2 \sqrt {4-2 \sqrt {3}}\right )}{x^4-2 \sqrt {2} x^2+2 \left (1+\sqrt {3}\right )}dx^2}{8 \sqrt {3}}+\frac {\sqrt {\frac {3}{2}} \left (\sqrt {2} \left (1+\sqrt {3}-\sqrt {12-6 \sqrt {3}}\right )-\left (1+\sqrt {3}-\sqrt {12-6 \sqrt {3}}\right ) x^2\right )}{2 \left (x^4-2 \sqrt {2} x^2+2 \left (1+\sqrt {3}\right )\right )}}{\sqrt {3}}\right )}{3-\sqrt {3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {3} \left (\frac {\left (3 \sqrt {2}-\sqrt {6}-2 \sqrt {6-3 \sqrt {3}}\right ) x^2+2 \left (3 \sqrt {4-2 \sqrt {3}}-\sqrt {3} \left (2-\sqrt {4-2 \sqrt {3}}\right )\right )}{\sqrt {3} \left (x^4-2 \sqrt {2} x^2+2 \left (1+\sqrt {3}\right )\right )^2}-\frac {\sqrt {2} \left (1+\sqrt {3}-\sqrt {12-6 \sqrt {3}}\right )-\left (1+\sqrt {3}-\sqrt {12-6 \sqrt {3}}\right ) x^2}{2 \sqrt {2} \left (x^4-2 \sqrt {2} x^2+2 \left (1+\sqrt {3}\right )\right )}\right )}{3-\sqrt {3}}\) |
Input:
Int[(2*Sqrt[3]*x*(-8*Sqrt[6] + 16*Sqrt[2 - Sqrt[3]] + 8*Sqrt[3*(2 - Sqrt[3 ])] - 8*Sqrt[2*(2 - Sqrt[3])]*x^2 - 6*Sqrt[2]*x^4 + 2*Sqrt[6]*x^4 + 12*Sqr t[2 - Sqrt[3]]*x^4 + 4*Sqrt[3*(2 - Sqrt[3])]*x^4 - 4*Sqrt[2*(2 - Sqrt[3])] *x^6 + Sqrt[2 - Sqrt[3]]*x^8))/((2 + 2*Sqrt[3] - 2*Sqrt[2]*x^2 + x^4)^2*(- 4*Sqrt[3] + 4*Sqrt[3*(2 - Sqrt[3])]*x^2 - 3*x^4 + Sqrt[3]*x^4)),x]
Output:
(Sqrt[3]*((2*(3*Sqrt[4 - 2*Sqrt[3]] - Sqrt[3]*(2 - Sqrt[4 - 2*Sqrt[3]])) + (3*Sqrt[2] - Sqrt[6] - 2*Sqrt[6 - 3*Sqrt[3]])*x^2)/(Sqrt[3]*(2*(1 + Sqrt[ 3]) - 2*Sqrt[2]*x^2 + x^4)^2) - (Sqrt[2]*(1 + Sqrt[3] - Sqrt[12 - 6*Sqrt[3 ]]) - (1 + Sqrt[3] - Sqrt[12 - 6*Sqrt[3]])*x^2)/(2*Sqrt[2]*(2*(1 + Sqrt[3] ) - 2*Sqrt[2]*x^2 + x^4))))/(3 - Sqrt[3])
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Px_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.)*((d_) + (e_.)*(x_) + (f_ .)*(x_)^2)^(q_.), x_Symbol] :> Simp[(c/f)^p Int[Px*(d + e*x + f*x^2)^(p + q), x], x] /; FreeQ[{a, b, c, d, e, f, p, q}, x] && PolyQ[Px, x] && EqQ[c* d - a*f, 0] && EqQ[b*d - a*e, 0] && (IntegerQ[p] || GtQ[c/f, 0]) && ( !Inte gerQ[q] || LeafCount[d + e*x + f*x^2] <= LeafCount[a + b*x + c*x^2])
Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[P q, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 1]}, Simp[(b*f - 2*a*g + (2*c*f - b*g)*x)*((a + b*x + c*x^2)^ (p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)) Int [(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (2*p + 3)* (2*c*f - b*g), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^ 2 - 4*a*c, 0] && LtQ[p, -1]
Int[(u_)*(x_)^(m_.), x_Symbol] :> Simp[1/(m + 1) Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /; FreeQ[m, x] && NeQ[m, -1] && Function OfQ[x^(m + 1), u, x]
Time = 0.68 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.57
| method | result | size |
| risch | \(\frac {2 \sqrt {3}\, \left (\frac {\sqrt {2}\, \sqrt {3}\, x^{2}}{12}-\frac {\sqrt {3}}{6}\right )}{2+2 \sqrt {3}-2 \sqrt {2}\, x^{2}+x^{4}}\) | \(43\) |
| default | \(-\frac {\sqrt {3}\, \sqrt {2}\, \left (\left (\sqrt {3}-1\right ) x^{2}+\sqrt {2}-\sqrt {3}\, \sqrt {2}\right )}{2 \left (-3+\sqrt {3}\right ) \left (2+2 \sqrt {3}-2 \sqrt {2}\, x^{2}+x^{4}\right )}\) | \(57\) |
| parallelrisch | \(-\frac {\sqrt {3}\, \left (x^{8} \sqrt {2}\, \sqrt {6}\, \sqrt {3}+24-9 \sqrt {3}\, x^{8}+42 \sqrt {2}\, \sqrt {3}\, x^{6}-14 x^{6} \sqrt {6}\, \sqrt {3}+3 x^{8}+6 x^{6} \sqrt {2}-18 \sqrt {6}\, x^{6}-24 \sqrt {2}\, \sqrt {3}\, x^{2}-8 x^{2} \sqrt {6}\, \sqrt {3}+8 \sqrt {6}\, \sqrt {3}\, \sqrt {2}+96 \sqrt {2}\, x^{2}-48 \sqrt {6}\, x^{2}-24 \sqrt {3}\right )}{72 \left (-x^{4}+2 \sqrt {2}\, x^{2}-2 \sqrt {3}-2\right )^{2}}\) | \(148\) |
| gosper | \(-\frac {\left (-x^{2}+\sqrt {2}\right ) \sqrt {3}\, \left (\sqrt {2}\, x^{8}-\sqrt {6}\, x^{8}+8 \sqrt {3}\, x^{6}-8 x^{6}+12 \sqrt {2}\, x^{4}-12 \sqrt {6}\, x^{4}+16 \sqrt {3}\, x^{2}-16 x^{2}-8 \sqrt {2}+8 \sqrt {6}\right )}{2 \left (-x^{4}+2 \sqrt {2}\, x^{2}-2 \sqrt {3}-2\right ) \left (-x^{4}+2 \sqrt {2}\, x^{2}+2 \sqrt {3}-2\right ) \left (\sqrt {3}\, x^{4}-3 x^{4}+6 \sqrt {2}\, x^{2}-2 \sqrt {6}\, x^{2}-4 \sqrt {3}\right )}\) | \(163\) |
Input:
int(2*3^(1/2)*x*(-4*6^(1/2)+4*2^(1/2)-8*(3^(1/2)-1)*x^2-6*2^(1/2)*x^4+2*6^ (1/2)*x^4+12*(1/2*6^(1/2)-1/2*2^(1/2))*x^4+4*(3/2*2^(1/2)-1/2*6^(1/2))*x^4 -4*(3^(1/2)-1)*x^6+(1/2*6^(1/2)-1/2*2^(1/2))*x^8)/(2+2*3^(1/2)-2*2^(1/2)*x ^2+x^4)^2/(-4*3^(1/2)+4*(3/2*2^(1/2)-1/2*6^(1/2))*x^2-3*x^4+3^(1/2)*x^4),x ,method=_RETURNVERBOSE)
Output:
2*3^(1/2)*(1/12*2^(1/2)*3^(1/2)*x^2-1/6*3^(1/2))/(2+2*3^(1/2)-2*2^(1/2)*x^ 2+x^4)
Timed out. \[ \int \frac {2 \sqrt {3} x \left (-8 \sqrt {6}+16 \sqrt {2-\sqrt {3}}+8 \sqrt {3 \left (2-\sqrt {3}\right )}-8 \sqrt {2 \left (2-\sqrt {3}\right )} x^2-6 \sqrt {2} x^4+2 \sqrt {6} x^4+12 \sqrt {2-\sqrt {3}} x^4+4 \sqrt {3 \left (2-\sqrt {3}\right )} x^4-4 \sqrt {2 \left (2-\sqrt {3}\right )} x^6+\sqrt {2-\sqrt {3}} x^8\right )}{\left (2+2 \sqrt {3}-2 \sqrt {2} x^2+x^4\right )^2 \left (-4 \sqrt {3}+4 \sqrt {3 \left (2-\sqrt {3}\right )} x^2-3 x^4+\sqrt {3} x^4\right )} \, dx=\text {Timed out} \] Input:
integrate(2*3^(1/2)*x*(-4*6^(1/2)+4*2^(1/2)-8*(3^(1/2)-1)*x^2-6*2^(1/2)*x^ 4+2*6^(1/2)*x^4+12*(1/2*6^(1/2)-1/2*2^(1/2))*x^4+4*(3/2*2^(1/2)-1/2*6^(1/2 ))*x^4-4*(3^(1/2)-1)*x^6+(1/2*6^(1/2)-1/2*2^(1/2))*x^8)/(2+2*3^(1/2)-2*2^( 1/2)*x^2+x^4)^2/(-4*3^(1/2)+4*(3/2*2^(1/2)-1/2*6^(1/2))*x^2-3*x^4+3^(1/2)* x^4),x, algorithm="fricas")
Output:
Timed out
Time = 18.36 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.83 \[ \int \frac {2 \sqrt {3} x \left (-8 \sqrt {6}+16 \sqrt {2-\sqrt {3}}+8 \sqrt {3 \left (2-\sqrt {3}\right )}-8 \sqrt {2 \left (2-\sqrt {3}\right )} x^2-6 \sqrt {2} x^4+2 \sqrt {6} x^4+12 \sqrt {2-\sqrt {3}} x^4+4 \sqrt {3 \left (2-\sqrt {3}\right )} x^4-4 \sqrt {2 \left (2-\sqrt {3}\right )} x^6+\sqrt {2-\sqrt {3}} x^8\right )}{\left (2+2 \sqrt {3}-2 \sqrt {2} x^2+x^4\right )^2 \left (-4 \sqrt {3}+4 \sqrt {3 \left (2-\sqrt {3}\right )} x^2-3 x^4+\sqrt {3} x^4\right )} \, dx=\text {Too large to display} \] Input:
integrate(2*3**(1/2)*x*(-4*6**(1/2)+4*2**(1/2)-8*(3**(1/2)-1)*x**2-6*2**(1 /2)*x**4+2*6**(1/2)*x**4+12*(1/2*6**(1/2)-1/2*2**(1/2))*x**4+4*(3/2*2**(1/ 2)-1/2*6**(1/2))*x**4-4*(3**(1/2)-1)*x**6+(1/2*6**(1/2)-1/2*2**(1/2))*x**8 )/(2+2*3**(1/2)-2*2**(1/2)*x**2+x**4)**2/(-4*3**(1/2)+4*(3/2*2**(1/2)-1/2* 6**(1/2))*x**2-3*x**4+3**(1/2)*x**4),x)
Output:
-(x**2*(-54567305088918325173851069823287372660371022840022464733663230138 34002086560494644585837460250822552784182692025735315573748238551705640375 65486988312005865671634234779703889861953362813643197499390540916862725858 46757687337485231674401989261528405637315606216926966045264716293396592352 23383331687932479778626028846196746440625127681767883166364268384068762702 35122400178224887716549955749309172654293474389760498397384047729251494541 22074040366883850829324452942498933894171898568540165914669010674616900354 72610943591666237*sqrt(2) + 3150444828203943090284248911440571141311610235 57927923814788993224130403308803652390052013806250541842135615440095453307 29810338021537142641531454574510385463575917958185798198894457914728653307 11748390510068228739206974952443761825512734425571121684282060865386373072 16820207586829630712626609081512864977709977845141231382773808453476510528 83770100571040684361560756644358956214767189301969185262851364247808438684 78213287754621448169485372656127223315321009931780038233901403015539579995 172649170704644595639876157101524442*sqrt(6)) + 10913461017783665034770213 96465747453207420456800449294673264602766800417312098928917167492050164510 55683653840514706311474964771034112807513097397662401173134326846955940777 97239067256272863949987810818337254517169351537467497046334880397852305681 12746312124338539320905294325867931847044676666337586495955725205769239349 28812502553635357663327285367681375254047024480035644977543309991149861...
Leaf count of result is larger than twice the leaf count of optimal. 554 vs. \(2 (57) = 114\).
Time = 0.28 (sec) , antiderivative size = 554, normalized size of antiderivative = 7.29 \[ \int \frac {2 \sqrt {3} x \left (-8 \sqrt {6}+16 \sqrt {2-\sqrt {3}}+8 \sqrt {3 \left (2-\sqrt {3}\right )}-8 \sqrt {2 \left (2-\sqrt {3}\right )} x^2-6 \sqrt {2} x^4+2 \sqrt {6} x^4+12 \sqrt {2-\sqrt {3}} x^4+4 \sqrt {3 \left (2-\sqrt {3}\right )} x^4-4 \sqrt {2 \left (2-\sqrt {3}\right )} x^6+\sqrt {2-\sqrt {3}} x^8\right )}{\left (2+2 \sqrt {3}-2 \sqrt {2} x^2+x^4\right )^2 \left (-4 \sqrt {3}+4 \sqrt {3 \left (2-\sqrt {3}\right )} x^2-3 x^4+\sqrt {3} x^4\right )} \, dx =\text {Too large to display} \] Input:
integrate(2*3^(1/2)*x*(-4*6^(1/2)+4*2^(1/2)-8*(3^(1/2)-1)*x^2-6*2^(1/2)*x^ 4+2*6^(1/2)*x^4+12*(1/2*6^(1/2)-1/2*2^(1/2))*x^4+4*(3/2*2^(1/2)-1/2*6^(1/2 ))*x^4-4*(3^(1/2)-1)*x^6+(1/2*6^(1/2)-1/2*2^(1/2))*x^8)/(2+2*3^(1/2)-2*2^( 1/2)*x^2+x^4)^2/(-4*3^(1/2)+4*(3/2*2^(1/2)-1/2*6^(1/2))*x^2-3*x^4+3^(1/2)* x^4),x, algorithm="maxima")
Output:
-1/288*sqrt(3)*(2*3^(3/4)*sqrt(2)*(sqrt(6)*(31*sqrt(3) + 57) + sqrt(6)*(25 *sqrt(3) + 39) - 96*sqrt(3)*sqrt(2) - 168*sqrt(2))*arctan(1/6*3^(3/4)*sqrt (2)*(x^2 - sqrt(2)))/(sqrt(6)*(sqrt(3)*sqrt(2) + 2*sqrt(2)) - 4*sqrt(3) - 6) - 9*(sqrt(6)*(5*sqrt(3)*sqrt(2) + 9*sqrt(2)) + 3*sqrt(6)*(sqrt(3)*sqrt( 2) + 5*sqrt(2)) - 48*sqrt(3) - 48)*log(x^4*(sqrt(3) - 3) - 2*x^2*(sqrt(6) - 3*sqrt(2)) - 4*sqrt(3))/(sqrt(6)*(2*sqrt(3)*sqrt(2) + 3*sqrt(2)) - 6*sqr t(3) - 12) + 9*(sqrt(6)*(5*sqrt(3)*sqrt(2) + 9*sqrt(2)) + 3*sqrt(6)*(sqrt( 3)*sqrt(2) + 5*sqrt(2)) - 48*sqrt(3) - 48)*log(x^4 - 2*sqrt(2)*x^2 + 2*sqr t(3) + 2)/(sqrt(6)*(2*sqrt(3)*sqrt(2) + 3*sqrt(2)) - 6*sqrt(3) - 12) + 18* (sqrt(6)*(43*sqrt(3) + 63) + 2*sqrt(6)*(17*sqrt(3) + 39) + 3*sqrt(6)*(sqrt (3) + 1) - 144*sqrt(3)*sqrt(2) - 240*sqrt(2))*arctan((x^2*(sqrt(3) - 3) - sqrt(6) + 3*sqrt(2))/sqrt(6*sqrt(6)*sqrt(2) + 12*sqrt(3) - 36))/((sqrt(6)* (2*sqrt(3)*sqrt(2) + 3*sqrt(2)) - 6*sqrt(3) - 12)*sqrt(6*sqrt(6)*sqrt(2) + 12*sqrt(3) - 36)) - 96*((sqrt(6)*(sqrt(3) - 3) + 3*sqrt(3)*sqrt(2) - 3*sq rt(2))*x^2 + sqrt(6)*(5*sqrt(3)*sqrt(2) + 9*sqrt(2)) - 18*sqrt(3) - 30)/(( sqrt(6)*(sqrt(3)*sqrt(2) + sqrt(2)) - 2*sqrt(3) - 6)*x^4 - 4*(sqrt(6)*(sqr t(3) + 1) - sqrt(3)*sqrt(2) - 3*sqrt(2))*x^2 + 4*sqrt(6)*(sqrt(3)*sqrt(2) + 2*sqrt(2)) - 16*sqrt(3) - 24))
Exception generated. \[ \int \frac {2 \sqrt {3} x \left (-8 \sqrt {6}+16 \sqrt {2-\sqrt {3}}+8 \sqrt {3 \left (2-\sqrt {3}\right )}-8 \sqrt {2 \left (2-\sqrt {3}\right )} x^2-6 \sqrt {2} x^4+2 \sqrt {6} x^4+12 \sqrt {2-\sqrt {3}} x^4+4 \sqrt {3 \left (2-\sqrt {3}\right )} x^4-4 \sqrt {2 \left (2-\sqrt {3}\right )} x^6+\sqrt {2-\sqrt {3}} x^8\right )}{\left (2+2 \sqrt {3}-2 \sqrt {2} x^2+x^4\right )^2 \left (-4 \sqrt {3}+4 \sqrt {3 \left (2-\sqrt {3}\right )} x^2-3 x^4+\sqrt {3} x^4\right )} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(2*3^(1/2)*x*(-4*6^(1/2)+4*2^(1/2)-8*(3^(1/2)-1)*x^2-6*2^(1/2)*x^ 4+2*6^(1/2)*x^4+12*(1/2*6^(1/2)-1/2*2^(1/2))*x^4+4*(3/2*2^(1/2)-1/2*6^(1/2 ))*x^4-4*(3^(1/2)-1)*x^6+(1/2*6^(1/2)-1/2*2^(1/2))*x^8)/(2+2*3^(1/2)-2*2^( 1/2)*x^2+x^4)^2/(-4*3^(1/2)+4*(3/2*2^(1/2)-1/2*6^(1/2))*x^2-3*x^4+3^(1/2)* x^4),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT: *** Warning: increasing stack s ize to 4096000. *** Warning: increasing stack size to 4096000. *** W arning: i
Timed out. \[ \int \frac {2 \sqrt {3} x \left (-8 \sqrt {6}+16 \sqrt {2-\sqrt {3}}+8 \sqrt {3 \left (2-\sqrt {3}\right )}-8 \sqrt {2 \left (2-\sqrt {3}\right )} x^2-6 \sqrt {2} x^4+2 \sqrt {6} x^4+12 \sqrt {2-\sqrt {3}} x^4+4 \sqrt {3 \left (2-\sqrt {3}\right )} x^4-4 \sqrt {2 \left (2-\sqrt {3}\right )} x^6+\sqrt {2-\sqrt {3}} x^8\right )}{\left (2+2 \sqrt {3}-2 \sqrt {2} x^2+x^4\right )^2 \left (-4 \sqrt {3}+4 \sqrt {3 \left (2-\sqrt {3}\right )} x^2-3 x^4+\sqrt {3} x^4\right )} \, dx=\text {Hanged} \] Input:
int(-(2*3^(1/2)*x*(12*x^4*(2^(1/2)/2 - 6^(1/2)/2) - 4*x^4*((3*2^(1/2))/2 - 6^(1/2)/2) + x^8*(2^(1/2)/2 - 6^(1/2)/2) - 4*2^(1/2) + 4*6^(1/2) + 6*2^(1 /2)*x^4 - 2*6^(1/2)*x^4 + 8*x^2*(3^(1/2) - 1) + 4*x^6*(3^(1/2) - 1)))/((2* 3^(1/2) - 2*2^(1/2)*x^2 + x^4 + 2)^2*(4*x^2*((3*2^(1/2))/2 - 6^(1/2)/2) - 4*3^(1/2) + 3^(1/2)*x^4 - 3*x^4)),x)
Output:
\text{Hanged}
Time = 0.39 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.25 \[ \int \frac {2 \sqrt {3} x \left (-8 \sqrt {6}+16 \sqrt {2-\sqrt {3}}+8 \sqrt {3 \left (2-\sqrt {3}\right )}-8 \sqrt {2 \left (2-\sqrt {3}\right )} x^2-6 \sqrt {2} x^4+2 \sqrt {6} x^4+12 \sqrt {2-\sqrt {3}} x^4+4 \sqrt {3 \left (2-\sqrt {3}\right )} x^4-4 \sqrt {2 \left (2-\sqrt {3}\right )} x^6+\sqrt {2-\sqrt {3}} x^8\right )}{\left (2+2 \sqrt {3}-2 \sqrt {2} x^2+x^4\right )^2 \left (-4 \sqrt {3}+4 \sqrt {3 \left (2-\sqrt {3}\right )} x^2-3 x^4+\sqrt {3} x^4\right )} \, dx=\frac {-8 \sqrt {6}\, x^{10}-32 \sqrt {6}\, x^{6}+192 \sqrt {6}\, x^{2}-48 \sqrt {3}\, x^{8}+64 \sqrt {3}\, x^{4}-128 \sqrt {3}+4 \sqrt {2}\, x^{14}-24 \sqrt {2}\, x^{10}-320 \sqrt {2}\, x^{2}+x^{16}-48 x^{8}+64 x^{4}+192}{8 x^{16}-64 x^{12}-2560 x^{4}+512} \] Input:
int(2*3^(1/2)*x*(-4*6^(1/2)+4*2^(1/2)-8*(3^(1/2)-1)*x^2-6*2^(1/2)*x^4+2*6^ (1/2)*x^4+12*(1/2*6^(1/2)-1/2*2^(1/2))*x^4+4*(3/2*2^(1/2)-1/2*6^(1/2))*x^4 -4*(3^(1/2)-1)*x^6+(1/2*6^(1/2)-1/2*2^(1/2))*x^8)/(2+2*3^(1/2)-2*2^(1/2)*x ^2+x^4)^2/(-4*3^(1/2)+4*(3/2*2^(1/2)-1/2*6^(1/2))*x^2-3*x^4+3^(1/2)*x^4),x )
Output:
( - 8*sqrt(6)*x**10 - 32*sqrt(6)*x**6 + 192*sqrt(6)*x**2 - 48*sqrt(3)*x**8 + 64*sqrt(3)*x**4 - 128*sqrt(3) + 4*sqrt(2)*x**14 - 24*sqrt(2)*x**10 - 32 0*sqrt(2)*x**2 + x**16 - 48*x**8 + 64*x**4 + 192)/(8*(x**16 - 8*x**12 - 32 0*x**4 + 64))